Problem 33 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation. Let x represent the number. Use mental math to solve the equation. Then check your solution. 28 decreased by a number is 18.

Step-by-Step Solution

Answer

The number 'x' is 10.

Step 1: Translate sentence into a mathematical equation

The sentence says '28 decreased by a number is 18'. This translates to an equation as follows: 28 - x = 18, where 'x' represents the unknown number.

Step 2: Solve the equation using mental math

Now, solve the equation 28 - x = 18 for x. This can be rearranged as: x = 28 - 18.

Step 3: Calculate the value of x

By subtracting 18 from 28, we find that x = 10.

Step 4: Check the solution

Substitute x = 10 into the original equation: 28 - 10 = 18. Since the left side equals the right side, the solution is correct.

Key Concepts

Writing Equations

Translating words into mathematical expressions is a fundamental skill when working with algebraic equations. This involves recognizing the mathematical operations described by words and arranging them into a coherent equation.

For instance, the phrase '28 decreased by a number' tells us we have the number 28, and we are performing a subtraction operation on it. The term 'is' in algebra typically translates to '=', indicating an equivalence. Writing an equation from words allows us to transform real-world situations into solvable mathematical problems.

In the given exercise, '28 decreased by a number is 18' becomes the equation \( 28 - x = 18 \), with 'x' as our variable representing the unknown number.

Solving Equations

Solving equations is the process of finding the value of the unknown variable that makes the equation true. For simple equations, mental math can often be used to quickly find the solution.

The equation \( 28 - x = 18 \) can be rearranged to get \( x = 28 - 18 \). This is a straightforward subtraction problem where you can easily compute \( x = 10 \). Solving equations mentally involves doing these arithmetic operations in your head, which is both satisfying and can enhance your number sense.

Mental Math

Mental math refers to using your mind to solve mathematical problems without the use of calculators or paper and pencil. It's a valuable skill that improves critical thinking and enhances your ability to handle numbers effectively.

Some tips for mental math include:

• Memorize basic arithmetic—addition, subtraction, multiplication, and division tables.
• Break down complex problems into smaller, more manageable parts.
• Use number sense to simplify calculations, such as rounding numbers and then adjusting the result.
• Practice regularly to increase speed and accuracy.

Using mental math to solve \( x = 28 - 18 \) is a practical exercise to bolster this skill since it requires simple subtraction.

Checking Solutions in Algebra

Once an equation has been solved, it’s essential to verify that the solution is correct. Checking the solution involves substituting the value of the variable back into the original equation and ensuring the left and right sides of the equation are equal.

For the exercise \( 28 - x = 18 \), we found the solution \( x = 10 \). To check the solution, we substitute 10 back for 'x' to get \( 28 - 10 = 18 \). Since this is a true statement, our solution \( x = 10 \) is correct. This step is crucial as it confirms not only the solution but also the understanding of the equation.